pinned vertices do no stay pinned when animating a cloth
The cloth simulator actually uses vertex weights (1 = pinned and 0 = unpinned).

Your vertex group has been assigned with 0 weight, as can be seen by entering weight paint mode (blue = 0 weight):



Reassign your vertex group with the weight slider on 1 (or paint some weights in weight paint mode):



Now the vertex group will have a weight of 1 and should stay pinned.



In the cloth modifiers physics settings enable pinning and select the cape vertex group.